5-list-coloring planar graphs with distant precolored vertices

We answer positively the question of Albertson asking whether every planar graph can be $5$-list-colored even if it contains precolored vertices, as long as they are sufficiently far apart from each other. In order to prove this claim, we also give bounds on the sizes of graphs critical with respect to 5-list coloring. In particular, if G is a planar graph, H is a connected subgraph of G and L is an assignment of lists of colors to the vertices of G such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G contains a subgraph with O(|H|^2) vertices that is not L-colorable.

💡 Research Summary

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The paper addresses a question posed by Albertson: whether a planar graph remains 5‑list‑colorable when some vertices are pre‑colored (i.e., have lists of size 1), provided that any two such vertices are sufficiently far apart. The authors give an affirmative answer and, in the process, obtain quantitative bounds on the size of minimal non‑L‑colorable subgraphs (critical graphs) with respect to 5‑list coloring.

The main results are two theorems. Theorem 2 states that if a planar graph G is equipped with a list assignment L in which every vertex has a list of size at least 5 except for a set of vertices that have lists of size 1, and if the distance between any two vertices with list size 1 is at least 20 780, then G is L‑colorable. This distance bound is derived from a careful analysis of “M‑valid” list assignments (a technical condition ensuring that vertices at distance at most M from a pre‑colored vertex have full lists of size 5) together with Thomassen’s classic theorem that every planar graph is 5‑choosable when the outer face is pre‑colored in a suitable way.

Theorem 3 deals with the structure of obstructions. Let H be a connected subgraph of a planar graph G, and let L be a list assignment such that every vertex outside H has a list of size at least 5 (lists on H may be arbitrary). If G is not L‑colorable, then G contains a subgraph F with at most 8·|V(H)|² vertices that is also not L‑colorable. In other words, any minimal counterexample to 5‑list‑colorability must be of size quadratic in the size of the pre‑colored region. This result provides an explicit O(|H|²) bound on the size of critical graphs.

The proof strategy combines several sophisticated tools:

1. M‑valid list assignments – The authors define a notion of validity that controls the interaction between pre‑colored vertices (lists of size 1) and the rest of the graph. The parameter M is chosen large enough to guarantee that vertices within distance M of a pre‑colored vertex have full lists, which is crucial for applying Thomassen’s theorem locally.

2. Thomassen’s list‑coloring theorem – The paper repeatedly invokes Thomassen’s result that a planar graph with a pre‑colored edge on the outer face and lists of size at least 5 elsewhere is L‑colorable. This theorem serves as a base case for the inductive arguments.

3. Critical graphs and weight analysis – A graph G is called H‑critical (with respect to L) if G ≠ H, every proper subgraph containing H is L‑colorable, but G itself is not. The authors assign weights to vertices and faces (depending on list sizes and whether they belong to the outer boundary) and define the total weight ω(G). Lemma 8 shows that for an H‑critical graph that is not just H plus a single chord, the total weight satisfies a linear inequality in |H|. This inequality forces the graph to be “small” in a precise sense.

4. Iterative decomposition – Using Lemma 7, the authors locate either a chord of the outer cycle or a vertex with three neighbours on the outer cycle. They then split the graph along appropriate chords or cycles, creating a sequence of smaller H‑critical subgraphs G₀ ⊃ G₁ ⊃ … ⊃ G_k. At each step the outer cycle shortens or the weight drops, guaranteeing that the process terminates after at most |H| steps. The cumulative effect yields the quadratic bound in Theorem 3.

5. Relaxed vertices and Lemma 5 – The concept of a “relaxed” vertex (a vertex for which there exist two distinct colorings differing only on that vertex) is introduced to control the propagation of colorings through the decomposition. Lemma 5 shows that when the list assignment is M‑valid, no two vertices with list size 3 are adjacent, and the graph is L‑colorable. This lemma is the key to proving Theorem 2 from Theorem 3.

The paper also acknowledges subsequent work by Postle and Thomas, who improved the bound in Theorem 3 from quadratic to linear, and consequently reduced the required distance between pre‑colored vertices. Nonetheless, the present work is the first to give an explicit polynomial bound and to settle Albertson’s question in full generality.

In summary, the authors successfully extend Thomassen’s 5‑choosability theorem to the setting where some vertices are pre‑colored, provided the pre‑colored vertices are sufficiently far apart. They achieve this by developing a refined structural theory of critical planar graphs, introducing a weight‑based induction, and carefully managing list assignments through the notion of M‑validity. The resulting theorems not only answer a long‑standing open problem but also provide concrete quantitative estimates (distance ≥ 20 780, subgraph size ≤ 8·|H|²) that pave the way for further refinements and applications in list‑coloring and graph‑drawing problems.